The Calderón–Zygmund estimate ‖D²u‖_p ≤ C ‖Δu‖_p

For every dimension n and 1 < p < ∞, this file proves the Lᵖ estimate of the second derivatives by the Laplacian on ℝⁿ: there is a constant C with

‖D²u‖_{Lᵖ} ≤ C ‖Δu‖_{Lᵖ}

for every Schwartz function u with values in a complex Hilbert space (TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian). This is the whole-space estimate at the heart of the Calderón–Zygmund W^{2,p} theory of strong solutions. For p = 2 it follows from Plancherel's theorem. For n ≥ 2 and p ≠ 2 it is a genuine singular integral estimate, and it is false at the endpoints p = 1 and p = ∞. In dimension one it is elementary: D²u and Δu have the same pointwise norm ‖u''‖, so it holds for every p. A corollary gives the same estimate for real Schwartz functions (TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian_real).

Both estimates are stated on an arbitrary finite-dimensional real inner product space V with its volume measure. They are proved on ℝⁿ and transported to V along a linear isometry V ≃ ℝⁿ, which preserves volume, the pointwise norm of D²u, and the Laplacian (TauCeti.laplacian_comp_linearIsometryEquiv_right).

The argument for n ≥ 3

On Schwartz functions ∂_v ∂_w = T_{m_{v,w}} ∘ Δ, where T_{m_{v,w}} is the L² Fourier multiplier with symbol m_{v,w}(ξ) = ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² (SchwartzMap.toLp_lineDerivOp_lineDerivOp_eq_fourierMultiplierCLM). So it suffices to bound T_{m_{v,w}} in the Lᵖ norm, which is the Calderón–Zygmund theorem (ContinuousLinearMap.exists_eLpNorm_le_of_norm_sub_le_mul_rpow) applied with the kernel k(x, y) = -D²Gₙ(x - y)(v, w) built from the Newtonian kernel Gₙ. Its three hypotheses are:

• the kernel representation of T_{m_{v,w}} off the support of its argument (TauCeti.ae_fourierMultiplierCLM_hessianSymbolLp_eq_neg_integral, which needs n ≥ 3);
• the size estimate ‖D²Gₙ(z)‖ ≲ ‖z‖⁻ⁿ (TauCeti.norm_iteratedFDeriv_two_newtonianKernel_le);
• the smoothness estimate ‖D²Gₙ(a) - D²Gₙ(b)‖ ≲ ‖a - b‖ ‖b‖⁻ⁿ⁻¹ for 2 ‖a - b‖ ≤ ‖b‖ (TauCeti.norm_iteratedFDeriv_two_newtonianKernel_sub_le), giving exponent δ = 1.

The theorem also asks for the same of the adjoint. The symbol is real, so T_{m_{v,w}} is self-adjoint (MeasureTheory.Lp.adjoint_fourierMultiplierCLM), also for the real inner product of L² that the Calderón–Zygmund theorem uses, and the adjoint has the same kernel.

Descent to low dimensions

The estimate in ℝⁿ⁺¹ implies the estimate in ℝⁿ with the same constant, by the method of descent. Write ℝⁿ⁺¹ = ℝ × ℝⁿ (EuclideanSpace.consEquivL) and test the estimate in ℝⁿ⁺¹ against the tensor product U(t, x) = ψ(t / R) u(x) (SchwartzMap.prodSMul) of a fixed nonzero profile ψ with u. For v, w ∈ ℝⁿ, ∂_{(0, v)} ∂_{(0, w)} U = ψ(· / R) ⊗ ∂_v ∂_w u and ΔU = ψ(· / R) ⊗ Δu + R⁻² ψ''(· / R) ⊗ u. The Lᵖ norm of a tensor product is the product of the norms (TauCeti.eLpNorm_prod_smul), and the dilation ψ(· / R) multiplies ‖ψ‖_p and ‖ψ''‖_p by the same factor (MeasureTheory.Measure.eLpNorm_comp_inv_smul). Dividing by it and letting R → ∞ leaves the estimate in ℝⁿ. This gives the planar case n = 2, where the Newtonian kernel is logarithmic and the kernel representation above is not available, and the case n = 1.

The full Hessian is controlled through an orthonormal basis (eᵢ): the norm of a bilinear map M is at most ∑ᵢⱼ ‖M(eᵢ, eⱼ)‖ (ContinuousMultilinearMap.opNorm_le_sum_norm_apply_orthonormalBasis), so ‖D²u‖_p ≤ ∑ᵢⱼ ‖∂_{eᵢ} ∂_{eⱼ} u‖_p.

Main declarations

• TauCeti.exists_eLpNorm_fourierMultiplierCLM_hessianSymbolLp_le: for n ≥ 3, the multiplier T_{m_{v,w}}, that is ∂_v ∂_w Δ⁻¹, is bounded in the Lᵖ norm for 1 < p < ∞.
• TauCeti.exists_eLpNorm_lineDerivOp_lineDerivOp_le_eLpNorm_laplacian: ‖∂_v ∂_w u‖_p ≤ C ‖Δu‖_p for Schwartz functions u, in every dimension.
• TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian: ‖D²u‖_p ≤ C ‖Δu‖_p for Schwartz functions u, and TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian_real for real ones, on every finite-dimensional real inner product space.

References

• E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter III, §1.3.
• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.9.

Descent from ℝⁿ⁺¹ to ℝⁿ

theorem TauCeti.exists_eLpNorm_fourierMultiplierCLM_hessianSymbolLp_le {n : ℕ} {F : Type u_1} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] (hn : 3 ≤ n) (v w : EuclideanSpace ℝ (Fin n)) {p : ENNReal} (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (f : ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)), MeasureTheory.eLpNorm (↑↑((MeasureTheory.Lp.fourierMultiplierCLM F (hessianSymbolLp v w)) f)) p MeasureTheory.volume ≤ ↑C * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund estimate for the Hessian multiplier. Let n ≥ 3, v, w ∈ ℝⁿ and 1 < p < ∞. The Fourier multiplier T_{m_{v,w}} on L²(ℝⁿ, F) with symbol ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖², that is the operator ∂_v ∂_w Δ⁻¹, is bounded in the Lᵖ norm: there is a constant C with ‖T_{m_{v,w}} f‖_p ≤ C ‖f‖_p for every f ∈ L².

It is the singular integral operator with kernel -D²Gₙ(x - y)(v, w), which satisfies the standard Calderón–Zygmund size and smoothness estimates.

theorem TauCeti.exists_eLpNorm_lineDerivOp_lineDerivOp_le_eLpNorm_laplacian {n : ℕ} {F : Type u_1} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] (v w : EuclideanSpace ℝ (Fin n)) {p : ENNReal} (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : SchwartzMap (EuclideanSpace ℝ (Fin n)) F), MeasureTheory.eLpNorm (⇑(LineDeriv.lineDerivOp v (LineDeriv.lineDerivOp w u))) p MeasureTheory.volume ≤ ↑C * MeasureTheory.eLpNorm (⇑(Laplacian.laplacian u)) p MeasureTheory.volume

The Calderón–Zygmund estimate for a second directional derivative. Let v, w ∈ ℝⁿ and 1 < p < ∞. There is a constant C with

‖∂_v ∂_w u‖_{Lᵖ} ≤ C ‖Δu‖_{Lᵖ}

for every Schwartz function u on ℝⁿ with values in a complex Hilbert space.

theorem TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian {F : Type u_1} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {p : ENNReal} (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : SchwartzMap V F), MeasureTheory.eLpNorm (iteratedFDeriv ℝ 2 ⇑u) p MeasureTheory.volume ≤ ↑C * MeasureTheory.eLpNorm (⇑(Laplacian.laplacian u)) p MeasureTheory.volume

The Calderón–Zygmund estimate ‖D²u‖_p ≤ C ‖Δu‖_p. Let 1 < p < ∞, and let V be a finite-dimensional real inner product space with its volume measure. There is a constant C with

‖D²u‖_{Lᵖ} ≤ C ‖Δu‖_{Lᵖ}

for every Schwartz function u on V with values in a complex Hilbert space, where D²u is the second Fréchet derivative of u.

theorem TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian_real {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {p : ENNReal} (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : SchwartzMap V ℝ), MeasureTheory.eLpNorm (iteratedFDeriv ℝ 2 ⇑u) p MeasureTheory.volume ≤ ↑C * MeasureTheory.eLpNorm (⇑(Laplacian.laplacian u)) p MeasureTheory.volume

The Calderón–Zygmund estimate ‖D²u‖_p ≤ C ‖Δu‖_p for real functions. Let 1 < p < ∞, and let V be a finite-dimensional real inner product space with its volume measure. There is a constant C with ‖D²u‖_{Lᵖ} ≤ C ‖Δu‖_{Lᵖ} for every real Schwartz function u on V.